Measurable versions of the LS category on laminations

TL;DR

This paper introduces two new measurable versions of the LS category for laminations, extending classical concepts to measurable and measured contexts, and establishing their fundamental properties.

Contribution

It defines and analyzes two measurable tangential categories for laminations, including a measure-dependent version, and proves their key properties.

Findings

Both categories are homotopy invariant.

They satisfy dimensional and cohomological bounds.

The measured category depends on a transverse invariant measure.

Abstract

We give two new versions of the LS category for the set-up of measurable laminations defined by Berm\'udez. Both of these versions must be considered as "tangential categories". The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the measured category, depends on the choice of a transverse invariant measure. We show that both of these "tangential categories" satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function.